How many days are you staying? As a base, I like the extra added space generally offered by an apartment and the possibility of being able to cook a meal in if I got tired of eating out, which could be the case if you're traveling for longer than a week overall. But as Uomo Grasso has pointed out, there are so many excellent restaurants in Florence, as well as cafes with pastries, that even a week stay could offer different places to eat at for every single meal.
I'd likely choose to eat out most of the time myself, so stay at an apartment and eat out!
The main difference would essentially be: do you want the convenience of the breakfast being offered by the hotel or would prefer to head out and find a spot for breakfast as well? There are several cafes that are organized to also offer savory options, included omelets, bagels, pancakes and whatnot as well as just grab a coffee and a pastry, depends on what you prefer to start your day with. If neither applies, you could also stop at a grocery store and get some basics and have breakfast at your apartment. I think this is where the main difference would be.